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Bounded C-star Dyson response for finite-range spin dynamics

Statement

Use $Γ = Z^{d}$ , $d \geq 1$ , finite on-site dimension $q \geq 1$ , and a uniformly bounded finite-range interaction $Φ$ with parameters $M \geq 1$ , $R < \infty$ , $κ < \infty$ , and $ℏ > 0$ as in Uniformly bounded finite-range quantum spin interaction. Assume Countable Choice and the exact finite-matrix C $^{*}$ -identity condition supplied in Finite matrix C-star algebra and density-state duality. Let $A$ be the norm-completed quasi-local spin algebra of Norm-completed quasi-local quantum spin algebra, and let $α_{t}$ be the strongly continuous automorphism group of Strongly continuous thermodynamic spin automorphism group. Let ${ρ_{Λ}}$ be consistent positive trace-one local density matrices; write $ϕ$ for their unique state extension supplied by Consistent local densities extend to an algebraic state, and additionally assume $ϕ \circ α_{t} = ϕ$ for every $t \in R$ . This is an algebraic state on $A$ , not a trace-class density on an infinite tensor-product Hilbert space.

For $A \in A$ , $B = B^{*} \in A$ , continuous real $h : [0, T] \to R$ , finite $T > 0$ , and dimensionless $ϵ \in R$ , put $B_{t} = α_{t} (B)$ and let $W_{ϵ}$ solve $W_{ϵ}^{'} (t) = \frac{i ϵ h (t)}{ℏ} B_{t} W_{ϵ} (t), W_{ϵ} (0) = I .$ The C $^{*}$ -algebra Dyson solution is unitary. Define $α_{t}^{ϵ} (A) = W_{ϵ} (t)^{*} α_{t} (A) W_{ϵ} (t)$ , $R_{A B} (u) = \frac{i}{ℏ} ϕ ([α_{u} (A), B]), C_{T} = \frac{2}{ℏ} \int_{0}^{T} ∣ h (s) ∣ ∥ B ∥ d s .$ Then $ϕ (α_{t}^{ϵ} (A)) = ϕ (α_{t} (A)) + ϵ \int_{0}^{t} h (s) R_{A B} (t - s) d s + r_{ϵ} (t), ∣ r_{ϵ} (t) ∣ \leq ∥ A ∥ \frac{(∣ ϵ ∣ C_{T})^{2} e^{∣ ϵ ∣ C_{T}}}{2} .$ When energy is the interpretation of $H_{Λ}$ and $h B$ , their units are J, $t$ is s, $ℏ$ is J s, and $C_{T}$ is dimensionless. No global trace Hamiltonian, spatial-to-temporal decay implication, or dc coefficient follows.

Facts & Assumptions

Given: Countable Choice, the finite local spin system and interaction, the finite-matrix identity, the completed quasi-local algebra and group, consistent local densities, an invariant extension $ϕ$ , bounded $A, B$ , continuous $h$ , and finite $T$ .

[A1]

The Axiom of Countable Choice ( ${A C}_{ω}$ ): Countable Choice is used for the norm-completion construction; no full Axiom of Choice is assumed here.

[F1]

Finite matrix C-star algebra and density-state duality: every nonzero finite local matrix algebra with its induced norm satisfies $∥ T^{*} T ∥ = ∥ T ∥^{2}$ .

[F2]

Uniformly bounded finite-range quantum spin interaction: $Φ (Z)$ is self-adjoint, vanishes for $∣ Z ∣ > M$ or diameter above $R$ , has per-site norm sum $κ$ , and defines the free-boundary finite-volume Hamiltonians.

[F3]

Norm-completed quasi-local quantum spin algebra: $A$ is the norm completion of the directed union of finite local matrix algebras, with no selected global tensor-product Hilbert space.

[F4]

Consistent local densities extend to an algebraic state: consistent positive trace-one local densities extend uniquely to a bounded normalized positive functional $ϕ$ on $A$ .

[F5]

Strongly continuous thermodynamic spin automorphism group: under [A1] and [F1]–[F3], the finite-volume dynamics converge to a strongly continuous one-parameter group $α_{t}$ of isometric $*$ -automorphisms.

[F6]

Banach-space Volterra and Dyson series with a norm remainder: on a unital C $^{*}$ -algebra, a norm-continuous skew-adjoint coefficient has a unique unitary Dyson solution; on a Banach space, the Volterra series has the stated norm remainder.

Proof

technique · apply the Volterra series to the dual Banach space of bounded functionals and identify its solution with unitary conjugation

Given: [A1], [F1]–[F6], and the invariant-state condition in the Statement.

1.1givenA1F1F2F3F4F5algebra

The finite-range group application uses Countable Choice [A1], the local matrix C $^{*}$ identity [F1], the finite-range interaction [F2], the quasi-local completion [F3], and the exact group conclusion [F5]. The extension $ϕ$ exists by [F4]. Since $t \mapsto α_{t} (B)$ is norm-continuous by [F5], $B_{t}$ is norm-continuous and $K_{t}^{*} (ψ) (X) : = \frac{i h (t)}{ℏ} ψ ([X, B_{t}])$ is a norm-continuous bounded operator on the continuous dual $A^{*}$ . Its norm is at most $2 ∣ h (t) ∣ ∥ B ∥ / ℏ$ , because automorphisms are isometric and $∥ [X, B_{t}] ∥ \leq 2 ∥ X ∥ ∥ B ∥$ . The dual $A^{*}$ is complete: a Cauchy sequence of bounded functionals has pointwise scalar limits, and the uniform Cauchy bound passes to that limit in the dual norm.

2.1givenF4F6step 1.1algebra

By [F6], the initial functional $ϕ$ has a unique Volterra solution $ψ_{ϵ} (t)$ in $A^{*}$ with first-order term $ϵ \int_{0}^{t} K_{s}^{*} (ϕ) d s$ and remainder dual norm at most $(∣ ϵ ∣ C_{T})^{2} e^{∣ ϵ ∣ C_{T}} / 2$ , since $∥ ϕ ∥ = 1$ . The C $^{*}$ -algebra clause of [F6] gives a norm-differentiable unitary $W_{ϵ}$ . Differentiating $X \mapsto ϕ (W_{ϵ} (t)^{*} X W_{ϵ} (t))$ shows this functional solves the same dual Volterra equation, so uniqueness gives $ψ_{ϵ} (t) (X) = ϕ (W_{ϵ} (t)^{*} X W_{ϵ} (t))$ .

3.1givenF5step 2.1algebra

Evaluate the first-order functional term at $X = α_{t} (A)$ . It is $\frac{i ϵ}{ℏ} \int_{0}^{t} h (s) ϕ ([α_{t} (A), B_{s}]) d s$ . The group law gives $[α_{t} (A), B_{s}] = α_{s} ([α_{t - s} (A), B])$ , and invariance of $ϕ$ removes $α_{s}$ . This is the stated convolution.

4.1givenF4step 2.1step 3.1algebra∎

Since $∥ α_{t} (A) ∥ = ∥ A ∥$ , evaluating the dual-norm remainder from step 2.1 at $α_{t} (A)$ gives the bound in the Statement. If $A = A^{*}$ as well as $B = B^{*}$ , positivity and Hermiticity of $ϕ$ make $R_{A B} (u)$ real.

Bounded C-star Dyson response for finite-range spin dynamics

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources